I know that a bijection is a surjection and an injection. Should I find the inverse of the function??? OR should I do element chasing to solve this problem?

Ah, I made a stupid mistake earlier today by accidentally saying all functions are injective, so I'll try to redeem myself answering this question.

1) Injection. Intuitively, we are proving that this function passes the "horizontal line test." I don't know if your professor prefers a certain method, but one approach is to take the derivative and show that f(x) is strictly decreasing.

2) Surjection. The image and the codomain are both the set of reals. This is evident since f(x) is defined everywhere and the limits as x approach negative and positive infinity are positive and negative infinity, respectively.